Problem: Let $g$ be a twice differentiable function, and let $g(4)=-2$, $g'(4)=0$, and $g''(4)=6$. What occurs in the graph of $g$ at the point $(4,-2)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(4,-2)$ is a minimum point. (Choice B) B $(4,-2)$ is a maximum point. (Choice C) C There's not enough information to tell.
Since $g'(4)=0$, we know that $x=4$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $g$ at this point according to these three cases: If $g''(4)>0$, the graph of $g$ has a minimum point at $x=4$. If $g''(4)<0$, the graph of $g$ has a maximum point at $x=4$. If $g''(4)=0$, the test is inconclusive. [Why is this so?] We are given that $g''(4)=6>0$. Therefore, $(4,-2)$ is a minimum point.